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Common diagonal solutions for second order complex family

Year 2019, Volume: 4 Issue: 1, 39 - 45, 16.05.2019

Abstract

In this paper for second order complex interval matrix family we consider the problem of existence and evaluation of common diagonal solutions for the Lyapunov matrix inequality. Necessary and sufficient conditions are given. Number of examples are given, where known sufficient conditions do not give common solutions.




References

  • E.~Kaszkurewicz, A.~Bhaya. Matrix Diagonal Stability in Systems and Computation. Birkhauser, Boston, 2000.
  • O.~Pastravanu, M.-H.~Matcovschi. Diagonal stability of interval matrices and applications. Linear Algebra and its Applications, 433(8):1646--1658, 2010.
  • H.K.~Wimmer. Diagonal matrix solutions of a discrete-time Lyapunov inequality. IEEE Transactions on Automatic Control, 43(3): 442--445, 1998.
  • C.R.~Johnson, T.D.~Lenker, S.K.~Narayan. Schur-type stability properties and complex diagonal scaling. Linear Algebra and its Applications , 429(10):2497--2520, 2008.
  • V.~Dzhafarov, T.~Büyükköroğlu. On the stability of a convex set of matrices. Linear Algebra and its Applications, 414(2): 547--559, 2006.
  • B.N.~Datta. Stability and inertia. Linear Algebra and its Applications, 302:563--600,1999.
  • G.W.~Cross. Three types of matrix stability. Linear Algebra and its Applications , 20(3):253--263, 1978.
  • G.~Barker, A.~Berman, R.~Plemmons. Positive diagonal solutions to the Lyapunov equations. Linear and Multilinear Algebra, 5(4): 249--256, 1978.
  • H.K.~Khalil. On the existence of positive diagonal $P$ such that $\uppercase{PA + A^T P} < 0$. IEEE Transactions on Automatic Control , 27(1): 181--184, 1982.
  • A.~Berman, D.~Hershkowitz. Matrix diagonal stability and its implications. SIAM Journal on Algebraic Discrete Methods, 4(3):377--382,1983.
  • D.~Hershkowitz, H.~Schneider. Lyapunov diagonal semistability of real $H$-matrices. Linear Algebra and its Applications, 71:119--149, 1985.
  • M.~Arcak, E.D.~Sontag. Diagonal stability of a class of cyclic systems and its connection with the secant criterion. Automatica , 42(9):1531--1537, 2006.
  • O.~Mason, R.~Shorten. On the simultaneous diagonal stability of a pair of positive linear systems. Linear Algebra and its Applications , 413(1):13--23, 2006.
  • B.~Yıldız T.~Büyükkoroğlu and V.~Dzhafarov. Common diagonal solutions to the Lyapunov inequalities for interval systems. Journal of Inequalities and Applications, 255:1--12, 2016.
  • O.~Pastravanu, M.-H.~Matcovschi. Sufficient conditions for Schur and Hurwitz diagonal stability of complex interval matrices. Linear Algebra and its Applications, 467:149--173, 2015.
  • T.~Büyükkoroğlu,Common diagonal Lyapunov function for third order linear switched system. Journal of Computational and Applied Mathematics, 236(15):3647--3653, 2012.}
Year 2019, Volume: 4 Issue: 1, 39 - 45, 16.05.2019

Abstract

References

  • E.~Kaszkurewicz, A.~Bhaya. Matrix Diagonal Stability in Systems and Computation. Birkhauser, Boston, 2000.
  • O.~Pastravanu, M.-H.~Matcovschi. Diagonal stability of interval matrices and applications. Linear Algebra and its Applications, 433(8):1646--1658, 2010.
  • H.K.~Wimmer. Diagonal matrix solutions of a discrete-time Lyapunov inequality. IEEE Transactions on Automatic Control, 43(3): 442--445, 1998.
  • C.R.~Johnson, T.D.~Lenker, S.K.~Narayan. Schur-type stability properties and complex diagonal scaling. Linear Algebra and its Applications , 429(10):2497--2520, 2008.
  • V.~Dzhafarov, T.~Büyükköroğlu. On the stability of a convex set of matrices. Linear Algebra and its Applications, 414(2): 547--559, 2006.
  • B.N.~Datta. Stability and inertia. Linear Algebra and its Applications, 302:563--600,1999.
  • G.W.~Cross. Three types of matrix stability. Linear Algebra and its Applications , 20(3):253--263, 1978.
  • G.~Barker, A.~Berman, R.~Plemmons. Positive diagonal solutions to the Lyapunov equations. Linear and Multilinear Algebra, 5(4): 249--256, 1978.
  • H.K.~Khalil. On the existence of positive diagonal $P$ such that $\uppercase{PA + A^T P} < 0$. IEEE Transactions on Automatic Control , 27(1): 181--184, 1982.
  • A.~Berman, D.~Hershkowitz. Matrix diagonal stability and its implications. SIAM Journal on Algebraic Discrete Methods, 4(3):377--382,1983.
  • D.~Hershkowitz, H.~Schneider. Lyapunov diagonal semistability of real $H$-matrices. Linear Algebra and its Applications, 71:119--149, 1985.
  • M.~Arcak, E.D.~Sontag. Diagonal stability of a class of cyclic systems and its connection with the secant criterion. Automatica , 42(9):1531--1537, 2006.
  • O.~Mason, R.~Shorten. On the simultaneous diagonal stability of a pair of positive linear systems. Linear Algebra and its Applications , 413(1):13--23, 2006.
  • B.~Yıldız T.~Büyükkoroğlu and V.~Dzhafarov. Common diagonal solutions to the Lyapunov inequalities for interval systems. Journal of Inequalities and Applications, 255:1--12, 2016.
  • O.~Pastravanu, M.-H.~Matcovschi. Sufficient conditions for Schur and Hurwitz diagonal stability of complex interval matrices. Linear Algebra and its Applications, 467:149--173, 2015.
  • T.~Büyükkoroğlu,Common diagonal Lyapunov function for third order linear switched system. Journal of Computational and Applied Mathematics, 236(15):3647--3653, 2012.}
There are 16 citations in total.

Details

Primary Language English
Journal Section Volume IV, Issue I, 2019
Authors

Bengi Yıldız

Publication Date May 16, 2019
Published in Issue Year 2019 Volume: 4 Issue: 1

Cite

APA Yıldız, B. (2019). Common diagonal solutions for second order complex family. Turkish Journal of Science, 4(1), 39-45.
AMA Yıldız B. Common diagonal solutions for second order complex family. TJOS. May 2019;4(1):39-45.
Chicago Yıldız, Bengi. “Common Diagonal Solutions for Second Order Complex Family”. Turkish Journal of Science 4, no. 1 (May 2019): 39-45.
EndNote Yıldız B (May 1, 2019) Common diagonal solutions for second order complex family. Turkish Journal of Science 4 1 39–45.
IEEE B. Yıldız, “Common diagonal solutions for second order complex family”, TJOS, vol. 4, no. 1, pp. 39–45, 2019.
ISNAD Yıldız, Bengi. “Common Diagonal Solutions for Second Order Complex Family”. Turkish Journal of Science 4/1 (May 2019), 39-45.
JAMA Yıldız B. Common diagonal solutions for second order complex family. TJOS. 2019;4:39–45.
MLA Yıldız, Bengi. “Common Diagonal Solutions for Second Order Complex Family”. Turkish Journal of Science, vol. 4, no. 1, 2019, pp. 39-45.
Vancouver Yıldız B. Common diagonal solutions for second order complex family. TJOS. 2019;4(1):39-45.